Abstract
We consider the Chromatic Sum Problem on bipartite graphs which appears to be much harder than the classical Chromatic Number Problem. We prove that the Chromatic Sum Problem is NP-complete on planar bipartite graphs with $\Delta \leq 5$, but polynomial on bipartite graphs with $\Delta \leq 3$, for which we construct an $O(n^{2})$-time algorithm. Hence, we tighten the borderline of intractability for this problem on bipartite graphs with bounded degree, namely: the case $\Delta =3$ is easy, $% \Delta =5$ is hard. Moreover, we construct a $27/26$-approximation algorithm for this problem thus improving the best known approximation ratio of $10/9$.
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Malafiejski, M., Giaro, K., Janczewski, R. et al. Sum Coloring of Bipartite Graphs with Bounded Degree. Algorithmica 40, 235–244 (2004). https://doi.org/10.1007/s00453-004-1111-4
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DOI: https://doi.org/10.1007/s00453-004-1111-4